Question: Suppose that $a$ and $b$ are nonzero integers such that two of the roots of
\[x^3 + ax^2 + bx + 9a\]coincide, and all three roots are integers.  Find $|ab|.$
Answer: Let the integer roots be $r,$ $r,$ and $s,$ so
\[x^3 + ax^2 + bx + 9a = (x - r)^2 (x - s).\]Expanding and matching coefficients, we get
\begin{align*}
2r + s  &= -a, \\
r^2 + 2rs &= b, \\
r^2 s &= -9a.
\end{align*}From the first and third equations, $r^2 s = 9(2r + s),$ so
\[s r^2 - 18r - 9s = 0.\]As a quadratic in $r,$ the discriminant is
\[\sqrt{18^2 - 4(s)(-9s)} = \sqrt{324 + 36s^2} = 3 \sqrt{s^2 + 9}.\]Since $r$ and $s$ are integers, $s^2 + 9$ must be a perfect square.  Let $s^2 + 9 = d^2,$ where $d > 0.$  Then
\[(d + s)(d - s) = 9.\]If $s = 0,$ then $a = 0,$ which is not allowed.  Otherwise, $d = \pm 5$ and $s = \pm 4.$  If $s = 4,$ then $r = 6,$ and $a = -16$ and $b = 84.$  If $s = -4,$ then $r = -6,$ and $a = 16$ and $b = 84.$  In either case,
\[|ab| = 16 \cdot 84 = \boxed{1344}.\]